= Free adjoint-scalar propagator in scalar-dependent gauge fixing
For <gauge fixing> with $F^a=\partial\cdot A^a+(n\cdot\partial\phi^a)/\sqrt2$, integrating out the <Nakanishi-Lautrup field> produces gauge-scalar mixing. The <Schur complement> of the free gauge-field kernel cancels the extra $(n\cdot p)^2/2$ in the scalar kernel, leaving $\langle\phi^a(p)\phi^b(-p)\rangle=\delta^{ab}/p^2$. This is independent of the gauge vector $n$, but still depends on the full momentum $p$.
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